Farmer taylor wants to fence a rectangular area of 1800 square feet and divided into 3 parts by fencing parallel to the shorter side. What is the minimum amount of fencing for this job?
draw the figure.
let s be the shorter side, L the longer side.
Area=sL
length of fencing= 2s+2L+2s=4s+2L
length of fencing=4s+ 2*Area/s
d(length)/ds=4-2A/s^2=0
s^2=2A/4
s= 1/2 sqrt2A=1/2 sqrt(3600)=30ft
L= 1800/30=60
To find the minimum amount of fencing required, we need to determine the dimensions of the rectangular area that will minimize the perimeter.
Let's assume the shorter side of the rectangle is 'x' feet. Since the rectangle is divided into 3 equal parts by fencing parallel to the shorter side, each part will have a width of 'x' feet.
The area of the rectangle is given as 1800 square feet, so we can set up the equation:
x * 3x = 1800
3x^2 = 1800
Dividing both sides by 3, we get:
x^2 = 600
Taking the square root of both sides, we have:
x = sqrt(600)
x ≈ 24.49 feet (rounded to two decimal places)
Therefore, the dimensions of the rectangle are approximately 24.49 feet by 3 * 24.49 = 73.47 feet.
To find the minimum amount of fencing required, we calculate the perimeter of the rectangle:
Perimeter = 2(length + width)
= 2(73.47 + 24.49)
= 2 * 97.96
= 195.92 feet (rounded to two decimal places)
So, the minimum amount of fencing required for the job is approximately 195.92 feet.